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REVIEW OF SCIENTIFIC INSTRUMENTS 76, 063112 共2005兲
Double-pass acousto-optic modulator system
E. A. Donley,a兲 T. P. Heavner, F. Levi,b兲 M. O. Tataw, and S. R. Jefferts
Time and Frequency Division, National Institute of Standards and Technology, 325 Broadway, Boulder,
Colorado 80305
共Received 2 March 2005; accepted 21 April 2005; published online 1 June 2005兲
A practical problem that arises when using acousto-optic modulators 共AOMs兲 to scan the laser
frequency is the dependence of the beam diffraction angle on the modulation frequency. Alignment
problems with AOM-modulated laser beams can be effectively eliminated by using the AOM in the
double-pass configuration, which compensates for beam deflections. On a second pass through the
AOM, the beam with its polarization rotated by 90° is deflected back such that it counterpropagates
the incident laser beam and it can be separated from the input beam with a polarizing beam splitter.
Here we present our design for a compact, stable, double-pass AOM with 75% double-pass
diffraction efficiency and a tuning bandwidth of 68 MHz full width at half maximum for light
transmitted through a single-mode fiber. The overall efficiency of the system 共defined as the optical
power out of the single-mode fiber divided by the optical power into the apparatus兲 is 60%.
关DOI: 10.1063/1.1930095兴
I. INTRODUCTION
For many laser cooling and trapping experiments, it is
necessary to shift and/or sweep the laser frequency at the end
of a cooling cycle. This is done to either adiabatically cool
the atoms to a lower temperature, or to manipulate them in
some other way, such as launching them in an atomic fountain. Most often, the laser frequency must be switched on
time scales shorter than 1 ms, and it may be that laser beams
with several different frequencies must be produced from a
single laser source.
Acousto-optic modulators 共AOMs兲 are widely used to
accomplish the frequency control in laser cooling experiments. When the laser frequency is scanned with an AOM,
the angle of the first-order diffracted beam shifts as well,
since the beam diffraction angle is a function of modulation
frequency. Changes in beam diffraction angle may be desirable for some applications where spatially resolved diffracted beams are needed, but for many applications any
change in the laser propagation direction is an unwanted side
effect. Using an AOM in the double-pass configuration is a
way to practically eliminate changes in beam steering during
frequency sweeps and jumps within the frequency tuning
bandwidth of the AOM. Here we present a detailed description of our modular double-pass system design and a study
of the system’s performance.
The article is organized as follows. In Sec. II, the laser
frequency jump and ramp sequence that is used to cool and
launch the atoms in the primary frequency standard at the
National Institute of Standards and Technology 共NIST兲 is
presented in detail. This provides a real-world example of the
level of laser frequency control that can be desirable for
a兲
Author to whom correspondence should be addressed; electronic mail:
[email protected]
b兲
Permanent address: Istituto Elettrotecnico Nazionale “G. Ferraris” Str.
Delle Cacce 91, Torino, Italy.
0034-6748/2005/76共6兲/063112/6/$22.50
atomic physics experiments. In Sec. III, background into the
properties of acousto-optic modulators is presented. In Sec.
IV, the double-pass configuration is introduced, our specific
design is discussed in detail, and the performance of our
design is presented.
II. ATOMIC FOUNTAIN LASER FREQUENCY
CONTROL
For laser cooling and trapping experiments, the optimum
laser intensity and detuning values that maximize the number
of laser-cooled atoms are not the same as the optimum values
for the lowest atom temperature. The laser intensity that
maximizes the number of atoms can depend on the details of
the atom source, but in general, a higher laser intensity is
better for a thermal source. The optimum laser detuning is
generally a few linewidths, but it can also depend on the
details of the atom source and the laser intensity. In contrast,
at low laser intensities the equilibrium atom temperature in
an optical molasses is equal to a small constant term
共⬃1 ␮K for 133Cs兲 plus a term that is proportional to the
laser intensity and inversely proportional to the laser
detuning.1,2 To simultaneously maximize the atom number
and minimize the atom temperature, often a brief “postcool”
stage is applied as the molasses is turned off. During the
postcool stage, the laser frequency is ramped further to the
red of the atomic resonance while the intensity is ramped to
zero.
The atom launch and postcool sequence for the vertical
laser beams used in the NIST-F1 atomic fountain clock at
NIST is shown in Fig. 1. NIST-F1 serves as the primary
frequency standard for the United States, and a detailed description of the apparatus has been published previously.3 We
accomplish the frequency sweeps and jumps shown in Fig. 1
with AOMs. The intensity ramp is produced by controlling
the rf power into the AOM with a voltage-controlled attenuator. With this sequence, we are able to collect about 107
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Rev. Sci. Instrum. 76, 063112 共2005兲
Donley et al.
FIG. 2. Momentum conservation diagrams for the absorption of a phonon.
The two cases 关共a兲 and 共b兲兴 represent two configurations for the incident and
diffracted light beams where the conditions are Bragg matched for a given
orientation of the phonon momentum. Note that ␬, and thus ␪i and ␪d, are
made large in the diagrams for clarity. The Bragg angle is normally about 1°
or less, and the phonon momentum is roughly two orders of magnitude
smaller than the photon momentum.
FIG. 1. Laser frequency and intensity ramps typically used for NIST-F1.
The intensity is shown on the left vertical scale in units of the saturation
intensity, Isat. The laser detuning from resonance is indicated on the right
vertical scale. The atoms are initially collected in an optical molasses for
⬃250 ms. To launch the atoms in a moving molasses, the frequencies of the
up 共down兲 beams are shifted from their initial frequencies by typically
+2.5 MHz 共−2.5 MHz兲 for 1 ms. Then the atoms are postcooled in the
moving frame for 2 ms by ramping down the laser intensity while linearly
shifting the frequencies of both beams further to the red by 44 MHz. Note
that the frequency and intensity curves for the horizontal molasses beams
are not shown. Those beams also have a postcool sequence, but there is no
frequency step when the atoms are launched.
atoms per cycle, cool them to ⬃0.5 ␮K, and launch them
into a microwave cavity structure for Ramsey interrogation.
III. ACOUSTO-OPTIC MODULATORS
A. Bragg scattering
For most practical applications of acousto-optic modulators, the Bragg description of the modulation process is a
good approximation to the behavior of the system.4,5 The
main features of AOMs are derived in this picture by treating
the modulation as a photon-phonon scattering process.
The Bragg matching condition can be derived by treating
the acoustic and optical fields as particles with momentum ␬
and k, respectively, where ␬ 共k兲 is the phonon 共photon兲 wave
vector for the acoustic 共optical兲 field; ␬ = ⍀ / vs, where ⍀ is
the rf modulation frequency and vs is the speed of sound in
the crystal. Similarly, k = ␻ / vL, where ␻ is the light frequency and vL is the speed of light in the crystal.
A scattering process between photons and phonons results in the absorption or emission of acoustic phonons. A
first-order scattering process between a photon and a single
phonon is described by the energy-momentum relations
␻d = ␻i ± ⍀,
kd = ki ± ␬ .
shows momentum conservation diagrams that describe
photon-phonon scattering events in which a phonon is absorbed 共+first-order diffraction兲. The photon momentum of
the diffracted light is equal to the sum of the momenta of the
phonon and the incident photon. Conservation of energy requires that the frequency of the diffracted beam be shifted
upward from ␻ to ␻ + ⍀ for a phonon absorption process; but
since ␻ Ⰷ ⍀, the frequency shift can be ignored in the momentum conservation analysis, and 兩ki兩 = 兩kd兩. Adding together
the x and y momentum components leads one to Bragg’s law
sin ␪B =
␬
,
2 · ki
共2兲
where ␪B is the Bragg angle, and ␪i = ␪d = ␪B. Note that we
have derived Eq. 共2兲 without considering the effect of the
boundaries of the acoustic medium. Often it is desirable to
calculate the Bragg angle outside of the crystal, ␪B,ext. To
close approximation, if the crystal boundaries are parallel to
␬, the external Bragg angle is larger than the internal Bragg
angle by a factor of n, where n is the refractive index of the
acoustic medium.
The cases shown in Figs. 2共a兲 and 2共b兲 represent the two
geometries where the Bragg matching condition is met for
+first-order phonon absorption for a given orientation of ␬.
Notice that the incident photon wave vector for one case and
the diffracted photon wave vector for the other case counterpropagate. This point is relevant to the discussion of the
double-pass configuration below. Diagrams similar to those
in Fig. 2 can be drawn for the single-phonon emission process, where the diffracted light enters the −first order, and
the frequency of the diffracted beam is shifted to ␻ − ⍀.
B. Acousto-optic devices
共1兲
The subscripts i and d designate whether the corresponding
photon is incident or diffracted. The sign depends on whether
the phonon is absorbed or emitted, which depends on the
relative orientations of the incident photon and phonon wave
vectors.
The Bragg matching condition that determines the optimum angles for the incident laser and acoustic beams for
peak first-order diffraction efficiency can be derived with
energy and momentum conservation arguments. Figure 2
A schematic drawing of an acousto-optic modulator is
shown in Fig. 3. A rf signal is fed to a strain transducer in
contact with the AOM crystal. The rf modulation at frequency ⍀ causes a traveling density wave to form inside the
crystal; the wave propagates at the speed of sound in the
crystal, vs, with the frequency ⍀. The refractive index is
therefore modulated with a wavelength of ⌳ = 2␲ · vs / ⍀, and
the crystal acts like a thick diffraction grating with the rulings traveling away from the transducer with a velocity vs.
The Bragg approximation is valid only when the acoustic
wave is describable by a plane wave and all phonons have
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063112-3
Double-pass AOM system
FIG. 3. 共Color online兲 Schematic drawing of an aligned AOM. The AOM is
shown in the mode where the light is diffracted into the +first order and the
laser frequency is up-shifted by the rf modulation frequency ⍀. Note that
phonons are absorbed in this configuration, and momentum is transferred to
the laser beam in the direction away from the strain transducer. The crystal
can also be aligned to be Bragg matched for the −first diffraction order 共not
shown兲. In that case, corresponding to a phonon emission process, all of the
laser beams are mirror symmetric about the horizontal axis with respect to
those shown in the figure, and the diffracted laser frequency is down shifted
by ⍀. Note that the external Bragg angle is larger than the internal Bragg
angle because of refraction at the crystal surfaces.
the same wave vector. In practice, this limiting case can be
achieved to a close approximation when the strain transducer
is long compared to the acoustic wavelength in the direction
of laser beam propagation and acoustic diffraction is minimized. It is therefore not uncommon to achieve greater than
80% diffraction efficiency into a single diffraction order with
careful engineering.5 The AOM that we use 共Crystal Technology, Inc. 3080-120兲6 has a TeO2 crystal driven with a
longitudinal acoustic mode. At a typical rf frequency of 80
MHz the external Bragg angle is 8 mrad for light with a
vacuum wavelength of 852 nm.
C. Bandwidth
The bandwidth is generally defined as the frequency
range over which the modulated laser power drops by 50%
共3 dB兲 from the peak value. For the AOM model that we use
the specified modulation bandwidth is 20 MHz. The diffraction efficiency that we measured versus rf modulation frequency is presented in Fig. 4 for a single pass through the
AOM of a 1.1 mm 关full width 共FW兲 1 / e2兴 collimated beam.
The full width at half maximum 共FWHM兲 of the curve is
⬃50 MHz, which is 2.5 times larger than the manufacturer’s
specified bandwidth. The diffraction efficiencies are the average of four measurements. The values for the absolute errors in the single-pass diffraction efficiency are up to 4%.
The double-pass measurements shown in Fig. 4 are discussed
in the next section.
IV. DOUBLE-PASS CONFIGURATION
We send light to our apparatus with a number of singlemode optical fibers that facilitate the delivery of laser light to
hard-to-reach points on the apparatus and act as spatial filters. While the spatial filtering improves the beam quality
and facilitates optics alignment downstream from a fiber, it
makes the coupling efficiency for light into the fiber ultrasensitive to the stability of the beam pointing upstream of the
Rev. Sci. Instrum. 76, 063112 共2005兲
FIG. 4. 共Color online兲 Measured single-pass and double-pass diffraction
efficiencies. When comparing the data in the graph, note that for the doublepass measurements the light was diffracted twice by the AOM, and the
measurements of the diffracted beam power were performed at the output of
a single-mode fiber, which had a coupling efficiency of 80%. It is the overall
diffraction efficiency through the fiber that is plotted in the figure for the
double-pass data. The double-pass diffraction efficiency at the fiber input is
75%. The double-pass diffraction efficiency measurements were performed
with and without the cat’s eye lens. Note that since the light frequency is
shifted twice for the double-pass measurements, the actual frequency tuning
bandwidth is twice the FWHM of the curves. For the double-pass measurements performed with the cat’s eye, this corresponds to 68 MHz.
fiber. This presents a challenge when the frequency of the
light being injected into the fiber is ramped with AOMs,
since the Bragg angle depends on the modulation frequency.
The alignment problems caused by AOM frequency
sweeps can be practically eliminated by using the doublepass configuration, in which the laser beam travels through
the AOM twice and the beam deflection is compensated in
the second pass. Modulating the light with the AOM twice
also means that the total frequency shift is twice the modulation frequency.
The two momentum conservation diagrams in Figs. 2共a兲
and 2共b兲 represent the momentum vectors for the two passes.
Notice that the incident and diffracted beams for one case
counterpropagate the diffracted and incident beams for
the other case. Thus with the alignment optimized, if the
+first-order diffracted beam is reflected back onto itself
through the AOM for a second pass, the +first-order diffracted beam for the second pass will spatially overlap the
original incident beam, independent of the modulation frequency.
Figure 5 is a schematic drawing of our double-pass
AOM apparatus, and a photograph is shown in Fig. 6. Light
with linear polarization oriented parallel to the page enters
the apparatus from the left as shown in Fig. 5. The beam is
steered through a polarizing beam splitter and a Galilean
telescope by two adjustable mirrors before it enters the
AOM. The Galilean telescope reduces the beam diameter
from 2.2 to 1.1 mm 共FW 1 / e2兲 and consists of a planoconvex lens and a plano-concave lens separated by 5 cm.
The position of the AOM is constrained by a 3 mm dowel
pin, around which the AOM can be rotated about an axis
through the center of the crystal. The AOM can be clamped
down to the mounting block with two screws after rough
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Rev. Sci. Instrum. 76, 063112 共2005兲
Donley et al.
needed to align the light polarization with the proper axis of
the fiber. The base plate to which the optics are mounted is
temperature controlled with an inexpensive commercial foil
heater and controller. We describe the function of some key
features in the apparatus in detail below.
A. Galilean telescope
FIG. 5. 共Color online兲 Schematic drawing of the double-pass system. M:
mirror; PBS: polarizing beam splitter; PCX: plano-convex lens; PCC: planoconcave lens; AOM: acousto-optic modulator; QWP: quarter-wave plate;
BB: beam block; HWP: half wave plate; FC: fiber coupler; PM: polarization
maintaining fiber. The circles that are attached to some of the optics represent angular adjustment knobs. The arrows above two of the lenses indicate
that they are held in lens-focusing mounts.
angular alignment of the AOM is achieved. Fine angular
alignment is accomplished by adjusting the orientation of the
two input mirrors. Immediately downstream from the AOM
is a quarter-wave plate that circularizes the polarization. The
plano-convex lens and mirror downstream from the AOM
combine to form the cat’s eye retroreflector, which is discussed in detail below. The zeroth-order undiffracted beam
is blocked with a beam block before the mirror. The
+first-order diffracted beam reflects back from the cat’s eye
and through the quarter-wave plate, where the polarization is
converted back to linear polarization such that the polarization vector is oriented perpendicular to the page on the second pass through the AOM. The light is modulated by the
AOM a second time and the resulting +first-order diffracted
beam for the second pass overlaps the incident laser beam.
The modulated light is then reflected from the adjustable
polarizing beam splitter and directed to an adjustable mirror,
a half wave plate, and a polarization-maintaining singlemode fiber coupler that contains an 11 mm coupling lens. If
beam misalignments of over 1 ␮m for the focused spot on
the fiber core are to be avoided, the angular tolerance for the
light beam leaving the AOM is ⬃0.2 mrad. Note that the
Galilean telescope reduces the angular steering of the beam
entering the fiber by a factor of 2. Alignment of the laser
beam into the fiber coupler is accomplished by adjusting the
angles of the mirror and the polarizing beam splitter. To
avoid fluctuations of the polarization of the light emerging
from the polarization-maintaining fiber, the half wave plate is
FIG. 6. 共Color online兲 Our double-pass apparatus. The dimensions of the
base plate are 25 cm⫻ 46 cm.
The diffraction efficiency for an AOM is a strong function of beam waist diameter, and increases with increasing
waist size before leveling off above 90% for a typical diameter of about 0.5 mm. The case enclosing the AOM that we
use has laser apertures of 3 mm in diameter, but the active
aperture over which the manufacturer guarantees that the device meets specifications is only 1 mm in diameter. The
beam size we chose is 1.1 mm 共FW 1 / e2兲 and is a trade-off
between performance and convenience.
The Gaussian beam waist diameter d for a plano-convex
lens may be estimated by the formula d = 4␭f / ␲D in terms of
the focal length f, the wavelength ␭, and the beam diameter
at the lens D. Following the manufacturer’s suggestion of
using a single plano-convex lens for focusing down the beam
while simultaneously optimizing the diffraction efficiency
would require the use of a lens with 1 m focal length. If
sufficient space is available on the optical table, then we can
follow this approach and achieve good diffraction efficiencies with a weakly focused beam.
We can dramatically reduce the size of the apparatus by
using a Galilean telescope to reduce the sizes of the laser
beams. In principle, by using lenses with very short focal
lengths, the length of the telescope can be reduced to a few
centimeters. However, there is a trade-off in how small to
make the focal lengths. In general, it is much more difficult
to fine tune the optical alignment in the apparatus when short
focal-length lenses are used to resize the beams. In our apparatus, the input beam has a diameter of 2.2 mm 共FW 1 / e2兲.
We reduce the diameter by a factor of 2 with a +100 mm
plano-convex and a −50 mm plano-concave lens spaced by
⬃50 mm such that the beam entering the AOM is collimated. We mounted the 100 mm converging lens on a commercial lens-focusing mount in order to fine tune the lens
spacing. Note that proper orientation of the planar side of the
lenses is required to minimize spherical aberrations4 共see Fig.
5兲. The alignment of a prototype of the apparatus was much
more difficult to optimize and maintain when the Galilean
telescope consisted of f = + 50 mm and f = −15 mm lenses.
We also measured the single-pass diffraction efficiency
of the AOM with a focused beam by removing the diverging
lens, without which the beam waist diameter was ⬃50 ␮m in
the AOM. If the incident laser beam is strongly focused into
the AOM, then there is effectively a range of angles of incidence for the incoming light, and only a fraction of the converging rays are Bragg matched with the crystal. From the
point of view of geometric optics, when the beam is focused
down, the light rays enter with an angular spread of
⬃20 mrad. This range is large compared to the external
Bragg angle 共⬃8 mrad兲, and in this mode only a component
of the zeroth-order beam is diffracted into the first-order
beam. Since the component of the zeroth-order beam that
diffracts depends on the rf frequency, a dark spot is observed
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063112-5
Rev. Sci. Instrum. 76, 063112 共2005兲
Double-pass AOM system
moving across the zeroth-order beam as the rf frequency is
swept over the frequency range of the AOM. The maximum
diffraction efficiency peaked at approximately 50% of what
we observed with a collimated beam.
B. Cat’s eye retroreflector
Having a cat’s eye7 as opposed to a simple flat-mirror
retroreflector dramatically improves the frequency-tuning
range 共bandwidth兲 for the double-pass apparatus. Owing to
the dependence of the Bragg angle on the rf modulation frequency, the angle of incidence onto the retroreflector depends on ⍀. This can cause the frequency-tuning bandwidth
for the double-pass apparatus to be much narrower than for
single-pass diffraction, since in practice, the optical alignment is optimized at only a single rf frequency. When the
beam arrives at the AOM for its second pass, an imperfect
retroreflector causes the angle of incidence on the AOM to
deviate from the external Bragg angle and the beam spot
position to shift on the AOM. This effect is most problematic
when a flat-mirror retroreflector is used.
The standard cat’s eye consists of a lens and a mirror
with their spacing equal to the focal length of the lens, f. For
our application, there is a special configuration for the cat’s
eye. For optimum performance the spacing between the
AOM and the lens should equal the lens focal length.
In this special configuration, as ⍀ is swept, the diffracted
angle is also swept, but when a lens is positioned a distance
f from the AOM, the rays emanate from the focal point of
the lens, and will emerge from the lens parallel to the zerothorder beam but deviated by a distance d = f · tan共2␪B,ext兲 for
the first-order beam. The angle of the mirror can then be
adjusted such that the diffracted beam always hits the mirror
at normal incidence independent of the modulation frequency.
The advantage of this special cat’s eye configuration was
previously noted by Boulanger and co-workers,8 but they did
not present their system in detail and did not inject the modulated light directly into a single-mode fiber. For our apparatus, we find that to efficiently inject the double-passed light
into the single-mode fiber without adjusting the beam collimation it is important that the lens-mirror spacing be close to
the lens focal length so that the beam is collimated at the
single-mode fiber coupler. To be able to optimize the fiber
coupling efficiency, we mounted the cat’s eye lens onto a
commercial lens focusing mount to optimize the lens-mirror
spacing. Also, in contrast to what is implied by Boulanger
and colleagues, the plano-convex lens in the cat’s eye does
not have to have a long focal length. We used a lens of half
the focal length that they recommended, found the alignment
sensitivity unchanged, and still achieved an ideal doublepass bandwidth.
Our cat’s eye retroreflector consists of a +75 mm focallength plano-convex lens and a planar mirror. While the performance is ideal when the AOM-lens spacing is equal to the
focal length, the spacing can be reduced by shifting the
whole cat’s eye toward the AOM to save space if desired
without dramatically compromising the bandwidth. For our
system, we observe only a 20% decrease in the frequency-
tuning bandwidth when the AOM-cat’s eye spacing is reduced by 50%.
In principle, we can achieve the same results using a
spherical mirror in place of the cat’s eye with the mirror
positioned at a distance equal to its radius of curvature from
the AOM. However, it is difficult to find an inexpensive concave mirror with high reflectivity.
C. Performance
We typically observe diffraction efficiencies exceeding
85% for a single pass and 75% for a double pass. Our data
are summarized in Fig. 4, where single-pass and double-pass
measurements are presented. It is important to emphasize
that, although the three curves are shown together, the values
of the diffraction efficiencies for the single-pass and doublepass measurements have different meanings. Whereas the
single-pass diffracted power was directly measured at the
AOM output, the light for the double-pass measurements
was diffracted by the AOM twice and this diffracted light
was injected into a single-mode fiber. The power for the
double-pass configurations was measured at the fiber output
to have a sensitive test on beam pointing stability. The peak
double-pass diffraction efficiency is 75% without the fiber, as
compared to 85% for the single-pass efficiency, which indicates that the second-pass diffraction through the AOM can
be more efficient than the first-pass diffraction.
The ratio of the FWHM for the single-pass measurements to the FWHM of the double-pass measurements performed with the cat’s eye is 1.43共7兲. Assuming a Gaussian
dependence, in theory, if there were no beam steering problems, the diffraction efficiency function for the single-pass
should be squared to arrive at the diffraction efficiency frequency dependence for the double pass. Thus, the optimum
ratio of the single-pass to the double-pass bandwidth is 冑2,
in agreement with what we observe. This agreement shows
that the cat’s eye acts as an ideal retroreflector. Note that the
frequency dependencies shown in Fig. 4 are in terms of the rf
modulation frequency, not the actual frequency shift, which
is two times larger for a double pass. So, in terms of the
frequency tuning range, the bandwidth of the double pass
with the cat’s eye is larger than that of the single pass by 冑2.
Quite striking is the difference between the double-pass
measurements performed with and without the cat’s eye lens.
Having the cat’s eye retroreflector as opposed to a planar
mirror increases the frequency-tuning bandwidth of the device by about one order of magnitude.
ACKNOWLEDGMENTS
The authors thank M. A. Lombardi and D. R. Smith for
helpful comments on this manuscript. This work is a contribution of NIST, an agency of the U.S. government, and is not
subject to copyright.
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063112-6
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Donley et al.
Products or companies named here are cited only in the interest of complete scientific description, and neither constitute nor imply endorsement
by NIST or by the US government. Other products may be found to serve
just as well.
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